The length of a rectangle is 1 meter more than twice its width. The… Step by Step

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The question

Question: The length of a rectangle is 1 meter more than twice its width.

The area is 36 square meters.

Find the length and the width.

1 Find what we know

What do we know?

The length is 2 times the width, plus 1.

The area is 36 square meters.

Area = length times width.

What do we want?

The length and the width.

2 Name the unknown

We know the length from the width.

So we call the width x.

x = width

The length is 2x + 1.

2x + 1 = length

Let's draw it.

The long side is 2x + 1.

The short side is x.

3 Write the equation

In words: width times length is the area.

x(2x + 1) = 36

So the equation is x(2x + 1) = 36.

There is x times x in it: a quadratic equation.

Now we solve it.

x(2x + 1) = 36

We open the bracket: 2x² + x = 36.

2x² + x = 36

4 Get 0 on one side

To solve it, we want 0 on the right side.

So we take away 36 on both sides.

We do it on both sides, so they stay equal.

2x² + x − 36 = 36 − 36

Now we have 2x² + x − 36 = 0.

2x² + x − 36 = 0

5 Find two numbers

There is a number in front of x², but it can still factor.

The number in front of x² is 2.

So we multiply 2 by −36: that is −72.

We need two numbers that multiply to −72 and add to 1.

2 × (−36) = −72

? × ? = −72, ? + ? = 1

They multiply to a minus number, so one number is plus and one is minus.

They add to a plus number, so the bigger one is plus.

× is − → one +, one −; bigger one +

We list the pairs that make 72, smallest first.

We take the smaller from the bigger.

We need 1.

8 and 9 give 1.

Yes!

With the signs: −8 and 9.

1 × 72: 72 − 1 = 71

2 × 36: 36 − 2 = 34

3 × 24: 24 − 3 = 21

4 × 18: 18 − 4 = 14

6 × 12: 12 − 6 = 6

8 × 9: 9 − 8 = 1 ✓

6 Split the middle

We write x as −8x + 9x.

It is the same amount.

2x² − 8x + 9x − 36 = 0

7 Group in pairs

We put the first two together, and the last two together.

(2x² − 8x) + (9x − 36) = 0

From the first pair we take out 2x.

From the second pair we take out 9.

Both leave the same bracket.

(2x² − 8x) = 2x(x − 4)

(9x − 36) = 9(x − 4)

Now both parts have x − 4 in them.

2x(x − 4) + 9(x − 4) = 0

8 Same bracket out

x − 4 is in both parts, so we take it out once.

What is left, 2x + 9, goes in the other bracket.

ab + cb = b(a + c)

So we get (x − 4)(2x + 9).

(x − 4)(2x + 9) = 0

9 One bracket is 0

Two numbers multiply to 0.

That only happens if one of them is 0.

So one bracket must be 0.

A × B = 0 → A = 0 or B = 0

We make each bracket 0.

Two small equations.

x − 4 = 0 or 2x + 9 = 0

In x − 4 = 0, we add 4 on both sides.

In 2x + 9 = 0, we take away 9 on both sides.

x − 4 + 4 = 0 + 4 or 2x + 9 − 9 = 0 − 9

Now we have x = 4 or 2x = −9.

x = 4 or 2x = −9

To remove times 2, we divide both sides by 2.

So x = 4 or x = −9/2.

x = 4 or x = −92

10 Answer the question

x = −9/2: a length cannot be negative.

So we throw it away.

x = −92 ✗

a length cannot be negative

Careful!

Both numbers make the equation true.

But only one makes sense in the story.

Always look back at the story.

Careful: x = −9/2 fits the equation, not the story

x = 4 works: a width.

x = 4 ✓

a width

The length is 2x + 1: 2 times 4 plus 1 is 9.

length = 2 × 4 + 1 = 9

The width is 4 m and the length is 9 m.

11 Check

Let's check in the story.

Length times width: 9 times 4 is 36, the area.

9 × 4 = 36

And 9 is 2 times 4 plus 1, like the story says.

So the answer is right!

9 = 2 × 4 + 1

Let's look back.

First: find what we know.

Then: name the unknown.

Then: write the equation.

Then: solve it.

Then: answer the question.

Last: check.

Answer